= Solution
Use target signature $(-,+,\ldots,+)$ and a closed-string spatial coordinate $0\le\sigma\le2\pi$. Write $Y$ for the compact coordinate and $X^A$ for the noncompact coordinates. In <conformal gauge>, the <wave equation> factorizes, so every coordinate is a sum of a function of $\tau+\sigma$ and a function of $\tau-\sigma$. The closed-string boundary conditions are
$$
X^A(\tau,\sigma+2\pi)=X^A(\tau,\sigma),\qquad Y(\tau,\sigma+2\pi)=Y(\tau,\sigma)+2\pi wR,\quad w\in\mathbb Z.
$$
Thus the <derivatives> of each chiral part are periodic and have <integer> <Fourier modes>; only their <worldsheet zero modes> can produce the winding displacement. Integrating these <Fourier series> gives the <closed-string mode expansion>
$$
X^A=x^A+\alpha'p^A\tau+i\sqrt{\frac{\alpha'}2}\sum_{m\ne0}\frac{\alpha_m^Ae^{-im(\tau-\sigma)}+\widetilde\alpha_m^Ae^{-im(\tau+\sigma)}}m,
$$
$$
Y=y+\alpha'\frac nR\tau+wR\sigma+i\sqrt{\frac{\alpha'}2}\sum_{m\ne0}\frac{\alpha_m^Ye^{-im(\tau-\sigma)}+\widetilde\alpha_m^Ye^{-im(\tau+\sigma)}}m.
$$
The oscillator reality conditions are $\alpha_m^\dagger=\alpha_{-m}$ and similarly for the tilded modes. The coefficient of $\tau$ follows by integrating the <canonical momentum> density $\dot Y/(2\pi\alpha')$. Single-valued wavefunctions of the center coordinate $y\sim y+2\pi R$ have <momentum> $n/R$, with $n\in\mathbb Z$. Equivalently, the compact chiral <worldsheet zero modes> are $\alpha'p_L(\tau+\sigma)/2$ and $\alpha'p_R(\tau-\sigma)/2$, where
$$
\boxed{p_L=\frac nR+\frac{wR}{\alpha'},\qquad p_R=\frac nR-\frac{wR}{\alpha'}.}
$$
Here tildes denote the left-moving sector. This convention will also fix the sign in <compact-circle closed-string level matching>.
Choose a noncompact light-cone pair and impose <light-cone gauge in string theory> $X^+=x^++\alpha'p^+\tau$, with $p^+\ne0$. The <Virasoro constraints> $(\partial_\pm X)^2=0$ then solve for $\partial_\pm X^-$ in terms of the transverse coordinates. For example, with $\partial_\pm=\partial_\tau\pm\partial_\sigma$, one has $\partial_\pm X^-=(\partial_\pm X^i)^2/(2\alpha'p^+)$. Thus there are two independent transverse oscillator <Fock spaces>, including $Y$ among the transverse coordinates. Quantization gives $[\alpha_m^i,\alpha_k^j]=m\delta^{ij}\delta_{m+k,0}$, and the same relation for the tilded oscillators, with the two sectors commuting. Lorentz consistency gives the <critical dimension of the bosonic string> $D=26$ and <string intercept> one in each sector. Accordingly
$$
N=\sum_{m>0}\alpha_{-m}^i\alpha_m^i,\qquad \widetilde N=\sum_{m>0}\widetilde\alpha_{-m}^i\widetilde\alpha_m^i
$$
count 24 transverse oscillator species. The two zero-mode constraints give the lower-dimensional <bosonic string mass spectrum>
$$
M^2=p_R^2+\frac4{\alpha'}(N-1)=p_L^2+\frac4{\alpha'}(\widetilde N-1).
$$
Taking their difference and average yields
$$
\boxed{\widetilde N-N+nw=0,\qquad M^2=\frac{n^2}{R^2}+\frac{w^2R^2}{\alpha'^2}+\frac2{\alpha'}(N+\widetilde N-2).}
$$
Every choice of transverse <string oscillators> and <integer> charges satisfying this <closed-string level matching> is a <physical string state> in light-cone gauge. The eliminated longitudinal oscillators do not supply additional polarizations.
To classify all massless states, positivity of the two compact terms implies $N+\widetilde N\le2$. When the sum is two, masslessness forces $n=w=0$, and <closed-string level matching> then forces $N=\widetilde N=1$. The states $\alpha_{-1}^i\widetilde\alpha_{-1}^j|0\rangle$ have $24^2=576$ polarizations. In 25 noncompact dimensions, their decomposition gives a <graviton>, a <Kalb–Ramond field>, a <dilaton>, two Abelian gauge vectors from the metric and two-form with one compact index, and the radius <scalar field>. Their physical polarization counts are
$$
275+253+2\cdot23+1+1=576.
$$
The first two numbers are the symmetric <traceless second-rank tensor> and <antisymmetric second-rank tensor> of the massless <little group> $SO(23)$.
When $N+\widetilde N=1$, <closed-string level matching> requires $|nw|=1$. The compact contribution obeys
$$
\frac{n^2}{R^2}+\frac{w^2R^2}{\alpha'^2}\ge\frac{2|nw|}{\alpha'}=\frac2{\alpha'},
$$
and equality occurs precisely at $R=\sqrt{\alpha'}$. The charges $n=w=\pm1$ have $(N,\widetilde N)=(1,0)$; the charges $n=-w=\pm1$ have $(N,\widetilde N)=(0,1)$. Each charge has 24 polarizations supplied by its single excited sector. At a general radius these have
$$
M^2=\left(\frac R{\alpha'}-\frac1R\right)^2.
$$
Finally, $N=\widetilde N=0$ requires $nw=0$. The <exceptional massless ground states of a bosonic circle> occur at
$$
R=\frac{|n|\sqrt{\alpha'}}2\quad(w=0,n\ne0),\qquad R=\frac{2\sqrt{\alpha'}}{|w|}\quad(n=0,w\ne0).
$$
Each charge is a <scalar field>. These special radii must be included if “general radius” is to mean a complete classification, rather than only a generic radius. \b[At a generic radius the massless spectrum is the neutral 576-state spectrum above; at exceptional radii one adds exactly the states just listed.] The neutral <ground state> remains tachyonic.
The map
$$
R'=\frac{\alpha'}R,\qquad (n',w')=(w,n)
$$
sends $p_L$ to $p_L$ and $p_R$ to $-p_R$. Reversing all right-moving compact <string oscillators>, $\alpha_m^Y\mapsto-\alpha_m^Y$, preserves their <commutators> and oscillator number; the left-moving oscillators are unchanged. Both the mass formula and <compact-circle closed-string level matching> are therefore unchanged, with a one-to-one map of the complete oscillator basis. \b[This proves <T-duality> of the full spectrum, including winding, massive and tachyonic states.]
At the <self-dual circle> the four single-oscillator charge families give $4\cdot24=96$ new massless states. Each family decomposes into a 25-dimensional gauge vector with 23 polarizations and one <scalar field>. There are also four massless <ground states>, with charges $(\pm2,0)$ and $(0,\pm2)$. Thus the <massless spectrum at the bosonic self-dual circle> has
$$
\boxed{576+96+4=676\ \text{polarizations}:\quad4\ \text{extra vectors and}\ 8\ \text{extra scalars}.}
$$
The gauge interpretation follows directly from the chiral compact-boson operators. With $\langle Y_L(z)Y_L(0)\rangle=-\alpha'\log z/2$, the exponential $e^{ikY_L}$ has <conformal weight> $\alpha'k^2/4$. At the self-dual radius,
$$
J_L^3=\frac{i}{\sqrt{\alpha'}}\partial Y_L,\qquad J_L^\pm=:e^{\pm2iY_L/\sqrt{\alpha'}}:
$$
have weight one and satisfy $J^3(z)J^3(0)\sim1/(2z^2)$, $J^3(z)J^\pm(0)\sim\pm J^\pm(0)/z$, and $J^+(z)J^-(0)\sim z^{-2}+2J^3(0)/z$. These are the level-one $SU(2)$ current relations; the right sector supplies another copy. Multiplying these currents by a noncompact oscillator produces six gauge vectors in total. The nine scalar <string vertex operators> $J_L^aJ_R^b$ form the $(3,3)$ representation, accounting for the radius <scalar field> and eight new <scalar fields>. The remaining neutral fields are the <graviton>, two-form and <dilaton>.
A <radius deformation as a Higgs mechanism> is the <expectation value> of the Cartan–Cartan scalar <string vertex operator> $J_L^3J_R^3\propto\partial Y_L\bar\partial Y_R$. Its stabilizer is $U(1)_L\times U(1)_R$, so
$$
\boxed{SU(2)_L\times SU(2)_R\longrightarrow U(1)_L\times U(1)_R,\qquad m_W=\left|\frac R{\alpha'}-\frac1R\right|.}
$$
The four compact single-oscillator <scalar field> polarizations become <longitudinal polarizations> of the four <massive vectors>, exactly as in the <Higgs mechanism>. The other four extra <scalar fields> have masses $4/R^2-4/\alpha'$ or $4R^2/\alpha'^2-4/\alpha'$ and need not remain massless; one pair becomes tachyonic on either side of the self-dual point. Together with the original bosonic <tachyon>, this means that the gauge-symmetry interpretation is formal and does not assert a stable bosonic-string vacuum.
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